$ F = \left[\begin{array}{rr}-2 & -2 \\ -1 & -1\end{array}\right]$ $ B = \left[\begin{array}{rr}5 & 3 \\ 4 & 2\end{array}\right]$ What is $ F B$ ?
Solution: Because $ F$ has dimensions $(2\times2)$ and $ B$ has dimensions $(2\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ F B = \left[\begin{array}{rr}{-2} & {-2} \\ {-1} & {-1}\end{array}\right] \left[\begin{array}{rr}{5} & \color{#DF0030}{3} \\ {4} & \color{#DF0030}{2}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ F$ , with the corresponding elements in column $j$ of the second matrix, $ B$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ F$ with the first element in ${\text{column }1}$ of $ B$ , then multiply the second element in ${\text{row }1}$ of $ F$ with the second element in ${\text{column }1}$ of $ B$ , and so on. Add the products together. $ \left[\begin{array}{rr}{-2}\cdot{5}+{-2}\cdot{4} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ F$ with the corresponding elements in ${\text{column }1}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{-2}\cdot{5}+{-2}\cdot{4} & ? \\ {-1}\cdot{5}+{-1}\cdot{4} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ F$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{-2}\cdot{5}+{-2}\cdot{4} & {-2}\cdot\color{#DF0030}{3}+{-2}\cdot\color{#DF0030}{2} \\ {-1}\cdot{5}+{-1}\cdot{4} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{-2}\cdot{5}+{-2}\cdot{4} & {-2}\cdot\color{#DF0030}{3}+{-2}\cdot\color{#DF0030}{2} \\ {-1}\cdot{5}+{-1}\cdot{4} & {-1}\cdot\color{#DF0030}{3}+{-1}\cdot\color{#DF0030}{2}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}-18 & -10 \\ -9 & -5\end{array}\right] $